General solution of the time evolution of two interacting harmonic oscillators

Bruschi, David Edward; Paraoanu, G. S.; Fuentes, Ivette; Wilhelm, Frank K.; Schell, Andreas W.

doi:10.48550/arxiv.1912.11087

Cited by 2 publications

(3 citation statements)

References 34 publications

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“…(83) This is indeed the Hamiltonian of two-coupled harmonic oscillators, with coupling −s1s2κ 2 c , but with the exotic feature that the kinetic terms can have negative signs. The analysis could then be carried on in the line of Bruschi et al (2019); Urzúa et al (2019); Moya-Cessa & Récamier (2020); Ramos-Prieto et al (2020) for example, to quote only recent studies.…”

Section: Coupled Harmonic Oscillator Formmentioning

confidence: 99%

An MHD spectral theory approach to Jeans' magnetized gravitational instability

Durrive,

Keppens,

Langer

2021

Preprint

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In this paper, we revisit the governing equations for linear magnetohydrodynamic (MHD) waves and instabilities existing within a magnetized, plane-parallel, selfgravitating slab. Our approach allows for fully non-uniformly magnetized slabs, which deviate from isothermal conditions, such that the well-known Alfvén and slow continuous spectra enter the description. We generalize modern MHD textbook treatments, by showing how self-gravity enters the MHD wave equation, beyond the frequently adopted Cowling approximation. This clarifies how Jeans' instability generalizes from hydro to magnetohydrodynamic conditions without assuming the usual Jeans' swindle approach. Our main contribution lies in reformulating the completely general governing wave equations in a number of mathematically equivalent forms, ranging from a coupled Sturm-Liouville formulation, to a Hamiltonian formulation linked to coupled harmonic oscillators, up to a convenient matrix differential form. The latter allows us to derive analytically the eigenfunctions of a magnetized, self-gravitating thin slab. In addition, as an example we give the exact closed form dispersion relations for the hydrodynamical p-and Jeans-unstable modes, with the latter demonstrating how the Cowling approximation modifies due to a proper treatment of self-gravity. The various reformulations of the MHD wave equation open up new avenues for future MHD spectral studies of instabilities as relevant for cosmic filament formation, which can e.g. use modern formal solution strategies tailored to solve coupled Sturm-Liouville or harmonic oscillator problems.

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Section: Coupled Harmonic Oscillator Formmentioning

confidence: 99%

An MHD spectral theory approach to Jeans' magnetized gravitational instability

Durrive,

Keppens,

Langer

2021

Preprint

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“…This fact has been shown numerically [1] and analytically [32] for g 1 = g 2 and D = 0. Attempts to generalize the previous results for g 1 = g 2 were made very recently in [33] where, through a relatively complicated procedure, an analytical result of the eigenfrequencies was obtained but only when D = 0. In the following, we show that, indeed, it is possible to get a complete analytical solution of the eigenvalues of ĤHopfield in the most general case, i.e., when g 1 = g 2 , D = 0 and the coupled system is off-resonance ω c = ω b .…”

Section: The Hopfield Modelmentioning

confidence: 99%

“…As expected, Eq. ( 2) contains, as a particular case, the results of [32] and [33] when, respectively, g 1 = g 2 or D = 0.…”

Section: The Hopfield Modelmentioning

confidence: 99%

Spectroscopy and critical quantum thermometry in the ultrastrong coupling regime

Salado-Mejía,

Román-Ancheyta,

Soto-Eguibar

et al. 2020

Preprint

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We present an exact analytical solution of the anisotropic Hopfield model, and we use it to investigate in detail the spectral and thermometric response of two ultrastrongly coupled quantum systems. Interestingly, we show that depending on the initial state of the coupled system, the vacuum Rabi splitting manifests significant asymmetries that may be considered spectral signatures of the counterintuitive decoupling effect.Using the coupled system as a thermometer for quantum thermodynamics applications, we obtain the ultimate bounds on the estimation of temperature that remain valid in the ultrastrong coupling regime. Remarkably, if the system performs a quantum phase transition, the quantum Fisher information exhibits periodic divergences, suggesting that one can have several points of arbitrarily high thermometry precision for such a critical quantum sensor.

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General solution of the time evolution of two interacting harmonic oscillators

Cited by 2 publications

References 34 publications

An MHD spectral theory approach to Jeans' magnetized gravitational instability

An MHD spectral theory approach to Jeans' magnetized gravitational instability

Spectroscopy and critical quantum thermometry in the ultrastrong coupling regime

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